In an arithmetic progression, the 2nd term is 8 and the 6th term is 20. What is the value of the 20th term of this A.P.?

Introduction / Context:
This question asks for a specific term in an arithmetic progression (A.P.) when two earlier terms are known. By using the general formula for the n-th term, we can set up equations to find the first term and common difference, and then compute the required term.

Given Data / Assumptions:
Let the first term of the A.P. be a and the common difference be d.The 2nd term T2 is 8.The 6th term T6 is 20.We are asked to find the 20th term T20.

Concept / Approach:
The n-th term of an A.P. is given by Tn = a + (n − 1)d. Using this formula for T2 and T6 gives us a pair of linear equations in a and d. Once we compute a and d, we can use the same formula to find T20.

Step-by-Step Solution:
Step 1: Write T2 and T6 using Tn = a + (n − 1)d.Step 2: T2 = a + d = 8.Step 3: T6 = a + 5d = 20.Step 4: Subtract the equation for T2 from that for T6: (a + 5d) − (a + d) = 20 − 8.Step 5: This simplifies to 4d = 12, so d = 3.Step 6: Substitute d = 3 into a + d = 8: a + 3 = 8, so a = 5.Step 7: Now find the 20th term: T20 = a + 19d = 5 + 19 × 3.Step 8: Compute 19 × 3 = 57, so T20 = 5 + 57 = 62.

Verification / Alternative check:
We can write several terms to confirm the pattern: 5, 8, 11, 14, 17, 20, ... The 2nd term is indeed 8 and the 6th is 20. With d = 3, every step increases by 3. Counting forward, the 20th term works out to be 62, matching our calculation.

Why Other Options Are Wrong:
Values like 56 or 65 might arise from miscounting the number of steps (using 18d or 20d instead of 19d) or from an incorrect d. The value 69 is too large and indicates an error in computing a or d. Only 62 is consistent with the given terms and the formula.

Common Pitfalls:
One frequent error is using Tn = a + nd instead of a + (n − 1)d, which shifts all positions by one. Another is rearranging the linear equations incorrectly when solving for a and d. Keeping track of indices carefully and checking earlier terms against the given values can catch such mistakes.

Final Answer:
The 20th term of the arithmetic progression is 62.